by Mike Martin with Fred Adler

In modeling biological phenomena, we have to consider another type of limiting process, called hysteresis. For instance, consider a situation when we apply an electrical current to stimulate a neuron. If we start with a low current or no current and increase the current, we find zero or no response from the neuron; however, when we pass through an input of nine units of current, the neuron begins then signaling with a Period of 1 unit and thereafter the neuron fires at a period that lie along the curve.

While in practice, we would find some noise and randomness in the measurements, consider an idealized situation with the following behavior:

We see that there is no response from the neuron up until time nine, and then there is a discontinuous jump from a value of zero to a value of 2.4. As the current is increased past 9 time units, the neuron responds by increasing the current, traveling on a smooth curve. So, with increasing current, we can summaraize what we find in with the following figure:

Now, if we start with the current very high (with a corresponding time value of 15), then we'll find a different behavior:

Interestingly, the turning down of the current has the neuron responding with oscillations that have a nonzero period up until a time value of 5 units; after 5 units, the neuron abruptly stops responding (period is zero). So, with decreasing current, we find a current that produces the following period response to the current:

Depending on whether or not we start with a zero current and increase or with a relatively high current and decrease, then we get different responses for the period of the neuron. This idea is called hysteresis and it is often in biological systems.